We prove that for each $\gamma \in (0,2)$, there is an exponent $d\gamma >2$, the ``fractal dimension of $\gamma$-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the $\gamma$-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $\gamma$-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that $d\gamma$ is a continuous, strictly increasing function of $\gamma$ and prove upper and lower bounds for $d\gamma$ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for $\gamma =2$ (which corresponds to spanning-tree weighted planar maps) our bounds give $3.4641\le d2\le 3.63299$ and in the limiting case we get $4.77485\le lim\gamma \to 2-d\gamma \le 4.89898$.